Optimal bandwidth selection for re-substitution entropy estimation

A new fusion approach of selecting an optimal bandwidth for re-substitution entropy estimator (RE) is presented in this study. When approximating the continuous entropy with density estimation, two types of errors will be generated: entropy estimation error (type-I error) and density estimation error (type-II error). These two errors are all strongly dependent on the undetermined bandwidths. Firstly, an experimental conclusion based on 24 typical probability distributions is demonstrated that there is some inconsistency between the optimal bandwidths associated with these two errors. Secondly, two different error measures for type-I and type-II errors are derived. A trade-off between type-I and type-II errors is a fundamental and potential property of our proposed method called REIþII . Thus, the fusion of these two errors is conducted and an optimal bandwidth for REIþII is solved. Finally, the experimental comparisons are carried out to verify the estimation performance of our proposed strategy. The discretization method is deemed to be the necessary preprocessing technology for the calculation of continuous entropy traditionally. So, the nine mostly used unsupervised discretization methods are introduced to give comparison of their computational performances with that of REIþII . And, five most popular estimators for entropy approximation are also plugged into our comparisons: splitting data estimator (SDE), cross-validation estimator (CVE), m-spacing estimator (mSE), mn-spacing estimator (mnSE), and nearest neighbor distance estimator (NNDE). The simulation studies on 24 different typical density distributions show that REIþII can obtain the better estimation performance among the involved methods. Meanwhile, the estimation behaviors of different entropy estimation methods are also revealed based on the comparative results. The empirical analysis demonstrates that REIþII is more insensitive to data and a better generalizable way for the estimation of continuous entropy. REIþII makes it possible for a handy optimal bandwidth to be derived from a given dataset. 2012 Elsevier Inc. All rights reserved.